So, could a cotton wig cap be damaging you hair?
The quick answer: no, not likely.
Cotton wig caps are okay if you are not sleeping in them.
Cotton surfaces are worse to sleep on because the full weight of your head is on the cotton cap or surface and all the tossing and turning adds to the abrasion (rubbing) and drying out of hair.
When you wear a wig cap during the day, however, your hair does not rub against the cap nearly as much. If your hair is in cornrows or flat twists that's even better because loose strands are much more vulnerable than hair that is bunched up together.
So overall, you are okay to wear a wig cap. Protect your hair by flat twisting or corn rowing it before putting the cap on.